Athletic put the Shrimpers under the cosh from the start of this Peter Butcher Memorial Trophy last-16 tie and in the 5th minute took the lead via a smart passing move.
Their second goal soon followed, with a cross that lofted over the head of goalkeeper Brad Cook into the far top corner of the net.
Newham were the masters of playing on their artificial surface and the torrential rain seemed to turn things even more to their advantage.
The Harwich defence were unable to cope with Athletic's two lightning-quick wingers and goals from Akijo Balliu (2), Harrison Tweddell (2) and Melvin Kambamba gave them a commanding lead at the break.
Despite the margin of the scoreline, Harwich also had opportunities but could not match Newham's clinical finishing.
A half-time talk by manager Matt Carmichael saw Harwich resolve to avoid a further drubbing and immediately Oakley McNair reduced the arrears.
On the hour Newham scored the best goal of the match with a spectacular 30-yard low drive that rocketed into the corner of the net.
However, Harwich were now giving a much better performance and Tom Watkins provided the assist for Louie Bloom (pictured) to score a well-taken second goal for the Shrimpers.
A positive for Harwich was the second-half, first-team debut of youngster Jeremiah Ajisafe, who impressed with an assured display at right-back.
Harwich team: Brad Cook, Oakley McNair, Teddy Johnson, Jack Wood, Reece Harte, Alfie Eldridge, Liam White, Gareth Heath, Zachary Pedro, Tiwa Odufuye and Louie Bloom. Substitutes: Jeremiah Ajisafe, Tom Watkins, Alfie Canning and Alex Pryke (all used).